If ABC × DEED = ABCABC; where A, B, C, D and E are different digits, what are the values of D and E ?

(a) D = 2, E = 0
(b) D = 0, E = 1
(c) D = 1, E = 0 ✓ UPSC's answer
(d) D = 1, E = 2

Why the answer is (c)

• The equation is $ABC \times DEED = ABCABC$. Since $ABC$ is a non-zero 3-digit number, we can divide both sides by $ABC$ to get $DEED / ABC = 1001$.

• This implies $DEED = ABC \times 1001$. However, a simpler algebraic manipulation is to note that $ABCABC = ABC \times 1001$. Thus, $ABC \times DEED = ABC \times 1001$.

• Dividing both sides by $ABC$ (which is non-zero), we find that $DEED = 1001$.

• The number $1001$ is a 4-digit number where the thousands digit is 1, the hundreds digit is 0, the tens digit is 0, and the units digit is 1.

• Comparing $DEED$ with $1001$, we identify $D = 1$ and $E = 0$.

• Therefore, the values are $D = 1$ and $E = 0$, which corresponds to option (c).

Why the other options are wrong

(a) D = 2, E = 0: Option (a) suggests $D=2, E=0$, which would make $DEED = 2002$, but $2002 \neq 1001$.
(b) D = 0, E = 1: Option (b) suggests $D=0, E=1$, which would make $DEED = 0110$ (or 110), but $110 \neq 1001$ and $D$ cannot be 0 for a 4-digit number representation in this context.
(d) D = 1, E = 2: Option (d) suggests $D=1, E=2$, which would make $DEED = 1221$, but $1221 \neq 1001$.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.